Theorem upfval 50228

Description: Function value of the class of universal properties. (Contributed by Zhi Wang, 24-Sep-2025.) (Proof shortened by Zhi Wang, 12-Nov-2025.)

Hypotheses

Ref	Expression
upfval.b	⊢ 𝐵 = (Base‘𝐷)
upfval.c	⊢ 𝐶 = (Base‘𝐸)
upfval.h	⊢ 𝐻 = (Hom ‘𝐷)
upfval.j	⊢ 𝐽 = (Hom ‘𝐸)
upfval.o	⊢ 𝑂 = (comp‘𝐸)

Assertion

Ref	Expression
upfval	⊢ (𝐷 UP 𝐸) = (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ 𝐶 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))})

Distinct variable groups:   𝐵,𝑓,𝑔,𝑘,𝑚,𝑤,𝑥,𝑦   𝐶,𝑓,𝑔,𝑘,𝑚,𝑤,𝑥,𝑦   𝐷,𝑓,𝑔,𝑘,𝑚,𝑤,𝑥,𝑦   𝑓,𝐸,𝑔,𝑘,𝑚,𝑤,𝑥,𝑦   𝑓,𝐻,𝑔,𝑘,𝑚,𝑤,𝑥,𝑦   𝑓,𝐽,𝑔,𝑘,𝑚,𝑤,𝑥,𝑦   𝑓,𝑂,𝑔,𝑘,𝑚,𝑤,𝑥,𝑦

Proof of Theorem upfval

Dummy variables 𝑏 𝑐 𝑑 𝑒 ℎ 𝑗 𝑜 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvexd 6892	. . . 4 ⊢ ((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) → (Base‘𝑑) ∈ V)
2		fveq2 6877	. . . . . 6 ⊢ (𝑑 = 𝐷 → (Base‘𝑑) = (Base‘𝐷))
3	2	adantr 486	. . . . 5 ⊢ ((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) → (Base‘𝑑) = (Base‘𝐷))
4		upfval.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐷)
5	3, 4	eqtr4di 2814	. . . 4 ⊢ ((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) → (Base‘𝑑) = 𝐵)
6		fvexd 6892	. . . . 5 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (Base‘𝑒) ∈ V)
7		simplr 781	. . . . . . 7 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → 𝑒 = 𝐸)
8	7	fveq2d 6881	. . . . . 6 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (Base‘𝑒) = (Base‘𝐸))
9		upfval.c	. . . . . 6 ⊢ 𝐶 = (Base‘𝐸)
10	8, 9	eqtr4di 2814	. . . . 5 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (Base‘𝑒) = 𝐶)
11		fvexd 6892	. . . . . 6 ⊢ ((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) → (Hom ‘𝑑) ∈ V)
12		simplll 787	. . . . . . . 8 ⊢ ((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) → 𝑑 = 𝐷)
13	12	fveq2d 6881	. . . . . . 7 ⊢ ((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) → (Hom ‘𝑑) = (Hom ‘𝐷))
14		upfval.h	. . . . . . 7 ⊢ 𝐻 = (Hom ‘𝐷)
15	13, 14	eqtr4di 2814	. . . . . 6 ⊢ ((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) → (Hom ‘𝑑) = 𝐻)
16		fvexd 6892	. . . . . . 7 ⊢ (((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) → (Hom ‘𝑒) ∈ V)
17		simp-4r 796	. . . . . . . . 9 ⊢ (((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) → 𝑒 = 𝐸)
18	17	fveq2d 6881	. . . . . . . 8 ⊢ (((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) → (Hom ‘𝑒) = (Hom ‘𝐸))
19		upfval.j	. . . . . . . 8 ⊢ 𝐽 = (Hom ‘𝐸)
20	18, 19	eqtr4di 2814	. . . . . . 7 ⊢ (((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) → (Hom ‘𝑒) = 𝐽)
21		fvexd 6892	. . . . . . . 8 ⊢ ((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) → (comp‘𝑒) ∈ V)
22		simp-5r 798	. . . . . . . . . 10 ⊢ ((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) → 𝑒 = 𝐸)
23	22	fveq2d 6881	. . . . . . . . 9 ⊢ ((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) → (comp‘𝑒) = (comp‘𝐸))
24		upfval.o	. . . . . . . . 9 ⊢ 𝑂 = (comp‘𝐸)
25	23, 24	eqtr4di 2814	. . . . . . . 8 ⊢ ((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) → (comp‘𝑒) = 𝑂)
26		simp-6l 799	. . . . . . . . . 10 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → 𝑑 = 𝐷)
27		simp-6r 800	. . . . . . . . . 10 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → 𝑒 = 𝐸)
28	26, 27	oveq12d 7430	. . . . . . . . 9 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (𝑑 Func 𝑒) = (𝐷 Func 𝐸))
29		simp-4r 796	. . . . . . . . 9 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → 𝑐 = 𝐶)
30		simp-5r 798	. . . . . . . . . . . . 13 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → 𝑏 = 𝐵)
31	30	eleq2d 2847	. . . . . . . . . . . 12 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (𝑥 ∈ 𝑏 ↔ 𝑥 ∈ 𝐵))
32		simplr 781	. . . . . . . . . . . . . 14 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → 𝑗 = 𝐽)
33	32	oveqd 7429	. . . . . . . . . . . . 13 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (𝑤𝑗((1st ‘𝑓)‘𝑥)) = (𝑤𝐽((1st ‘𝑓)‘𝑥)))
34	33	eleq2d 2847	. . . . . . . . . . . 12 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥)) ↔ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))))
35	31, 34	anbi12d 644	. . . . . . . . . . 11 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → ((𝑥 ∈ 𝑏 ∧ 𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥))) ↔ (𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥)))))
36	32	oveqd 7429	. . . . . . . . . . . . 13 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (𝑤𝑗((1st ‘𝑓)‘𝑦)) = (𝑤𝐽((1st ‘𝑓)‘𝑦)))
37		simplr 781	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) → ℎ = 𝐻)
38	37	oveqdr 7440	. . . . . . . . . . . . . 14 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (𝑥ℎ𝑦) = (𝑥𝐻𝑦))
39		simpr 490	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → 𝑜 = 𝑂)
40	39	oveqd 7429	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦)) = (⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦)))
41	40	oveqd 7429	. . . . . . . . . . . . . . 15 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚) = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))
42	41	eqeq2d 2772	. . . . . . . . . . . . . 14 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚) ↔ 𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚)))
43	38, 42	reueqbidv 3402	. . . . . . . . . . . . 13 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚) ↔ ∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚)))
44	36, 43	raleqbidv 3335	. . . . . . . . . . . 12 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚) ↔ ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚)))
45	30, 44	raleqbidv 3335	. . . . . . . . . . 11 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (∀𝑦 ∈ 𝑏 ∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚) ↔ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚)))
46	35, 45	anbi12d 644	. . . . . . . . . 10 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (((𝑥 ∈ 𝑏 ∧ 𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝑏 ∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚)) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))))
47	46	opabbidv 5171	. . . . . . . . 9 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝑏 ∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚))} = {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))})
48	28, 29, 47	mpoeq123dv 7487	. . . . . . . 8 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (𝑓 ∈ (𝑑 Func 𝑒), 𝑤 ∈ 𝑐 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝑏 ∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚))}) = (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ 𝐶 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))}))
49	21, 25, 48	csbied2 3884	. . . . . . 7 ⊢ ((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) → ⦋(comp‘𝑒) / 𝑜⦌(𝑓 ∈ (𝑑 Func 𝑒), 𝑤 ∈ 𝑐 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝑏 ∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚))}) = (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ 𝐶 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))}))
50	16, 20, 49	csbied2 3884	. . . . . 6 ⊢ (((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) → ⦋(Hom ‘𝑒) / 𝑗⦌⦋(comp‘𝑒) / 𝑜⦌(𝑓 ∈ (𝑑 Func 𝑒), 𝑤 ∈ 𝑐 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝑏 ∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚))}) = (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ 𝐶 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))}))
51	11, 15, 50	csbied2 3884	. . . . 5 ⊢ ((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) → ⦋(Hom ‘𝑑) / ℎ⦌⦋(Hom ‘𝑒) / 𝑗⦌⦋(comp‘𝑒) / 𝑜⦌(𝑓 ∈ (𝑑 Func 𝑒), 𝑤 ∈ 𝑐 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝑏 ∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚))}) = (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ 𝐶 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))}))
52	6, 10, 51	csbied2 3884	. . . 4 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → ⦋(Base‘𝑒) / 𝑐⦌⦋(Hom ‘𝑑) / ℎ⦌⦋(Hom ‘𝑒) / 𝑗⦌⦋(comp‘𝑒) / 𝑜⦌(𝑓 ∈ (𝑑 Func 𝑒), 𝑤 ∈ 𝑐 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝑏 ∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚))}) = (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ 𝐶 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))}))
53	1, 5, 52	csbied2 3884	. . 3 ⊢ ((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) → ⦋(Base‘𝑑) / 𝑏⦌⦋(Base‘𝑒) / 𝑐⦌⦋(Hom ‘𝑑) / ℎ⦌⦋(Hom ‘𝑒) / 𝑗⦌⦋(comp‘𝑒) / 𝑜⦌(𝑓 ∈ (𝑑 Func 𝑒), 𝑤 ∈ 𝑐 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝑏 ∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚))}) = (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ 𝐶 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))}))
54		df-up 50226	. . 3 ⊢ UP = (𝑑 ∈ V, 𝑒 ∈ V ↦ ⦋(Base‘𝑑) / 𝑏⦌⦋(Base‘𝑒) / 𝑐⦌⦋(Hom ‘𝑑) / ℎ⦌⦋(Hom ‘𝑒) / 𝑗⦌⦋(comp‘𝑒) / 𝑜⦌(𝑓 ∈ (𝑑 Func 𝑒), 𝑤 ∈ 𝑐 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝑏 ∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚))}))
55		ovex 7445	. . . 4 ⊢ (𝐷 Func 𝐸) ∈ V
56	9	fvexi 6891	. . . 4 ⊢ 𝐶 ∈ V
57	55, 56	mpoex 8081	. . 3 ⊢ (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ 𝐶 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))}) ∈ V
58	53, 54, 57	ovmpoa 7567	. 2 ⊢ ((𝐷 ∈ V ∧ 𝐸 ∈ V) → (𝐷 UP 𝐸) = (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ 𝐶 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))}))
59		reldmup 50227	. . . 4 ⊢ Rel dom UP
60	59	ovprc 7450	. . 3 ⊢ (¬ (𝐷 ∈ V ∧ 𝐸 ∈ V) → (𝐷 UP 𝐸) = ∅)
61		reldmfunc 50127	. . . . . 6 ⊢ Rel dom Func
62	61	ovprc 7450	. . . . 5 ⊢ (¬ (𝐷 ∈ V ∧ 𝐸 ∈ V) → (𝐷 Func 𝐸) = ∅)
63	62	orcd 887	. . . 4 ⊢ (¬ (𝐷 ∈ V ∧ 𝐸 ∈ V) → ((𝐷 Func 𝐸) = ∅ ∨ 𝐶 = ∅))
64		0mpo0 7495	. . . 4 ⊢ (((𝐷 Func 𝐸) = ∅ ∨ 𝐶 = ∅) → (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ 𝐶 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))}) = ∅)
65	63, 64	syl 18	. . 3 ⊢ (¬ (𝐷 ∈ V ∧ 𝐸 ∈ V) → (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ 𝐶 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))}) = ∅)
66	60, 65	eqtr4d 2799	. 2 ⊢ (¬ (𝐷 ∈ V ∧ 𝐸 ∈ V) → (𝐷 UP 𝐸) = (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ 𝐶 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))}))
67	58, 66	pm2.61i 184	1 ⊢ (𝐷 UP 𝐸) = (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ 𝐶 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))})

This proof depends on syntax axioms:  ¬ wn 3   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145  ∀wral 3077  ∃!wreu 3364  Vcvv 3451  ⦋csb 3847  ∅c0 4279  ⟨cop 4590  {copab 5167  ‘cfv 6531  (class class class)co 7412   ∈ cmpo 7414  1^st c1st 7988  2^nd c2nd 7989  Basecbs 17367  Hom chom 17419  compcco 17420   Func cfunc 18009   UP cup 50225

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7990  df-2nd 7991  df-func 18013  df-up 50226

This theorem is used by:  upfval2  50229  uppropd  50233  reldmup2  50234  relup  50235  uprcl  50236